Necessary condition and sufficient conditions for nonuniform wavelet frames in L2(K)

A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in L-2(R) was considered by Gabardo and Nashed [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal. 158 (1998) 209-241]. In this setting, the associated translation set is a spectrum. which is not necessarily a group nor a uniform discrete set, given Lambda = {0, r/N} + 2Z, where N >= 1 (an integer) and r is an odd integer with 1 <= r <= 2N - 1 such that r and N are relatively prime and Z is the set of all integers. The objective of this paper is to construct nonuniform wavelet frame on local fields. A necessary condition and four sufficient conditions for nonuniform wavelet frame on local fields are given.